Topic 10 Polynomial a + bx + cx 2 can be represented by the - - PDF document

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Fall 2008 Programming Development Techniques 1

Topic 10 Example: Symbolic Differentiation

Section 2.3.2 October 2008

Fall 2008 Programming Development Techniques 2

Symbolic differentiation

• Polynomial a + bx + cx2 can be represented by the

list (+ a (+ (* b x) (* c (* x x))))

• Derivative with respect to x is b + 2cx, which can be

represented by (+ b (* 2 (* c x)))

• How can the derivative be computed?

Fall 2008 Programming Development Techniques 3

Basic calculus (2 arguments only)

dy/dx = 0 if y is a constant or a variable other than x dx/dx = 1 d(u+ v)/dx = du/dx + dv/dx (NOTE Recursive!) d(u* v)/dx = u * dv/dx + v * du/dx (NOTE Recursive!)

Fall 2008 Programming Development Techniques 4

Want to build a procedure to do differentiation

• Think of the first rules as base conditions, then other

rules can be used to decompose a problem into something easier.

• What do we need to do to tell which rule is

applicable?

– Differentiate between a constant, variable (and what it is), product, and sum – Extract parts of an expression

Fall 2008 Programming Development Techniques 5

Data abstraction to the rescue!

Some constructors, selectors and predicates: (variable? x) (same-variable? x y) (sum? x) (product? x) (make-sum x y) (make-product x y) (sum-arg1 x) (product-arg1 x) (sum-arg2 x) (product-arg2 x)

Fall 2008 Programming Development Techniques 6

Now we can compute derivatives

; takes an expression and a variable and ; returns the derivitive of expr wrt var (define (deriv expr var) (cond ((number? expr) 0) ((variable? expr) (if (same-variable? expr var) 1 0)) ((sum? expr) (make-sum (deriv (sum-arg1 expr) var) (deriv (sum-arg2 expr) var)))

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Fall 2008 Programming Development Techniques 7

(deriv continued)

((product? expr) (make-sum (make-product (product-arg1 expr) (deriv (product-arg2 expr) var)) (make-product (product-arg2 expr) (deriv (product-arg1 expr) var)))) (else (error "Unknown type“ expr))))

Fall 2008 Programming Development Techniques 8

Implementation of lower layer

; takes an expression and returns #t ; if it is a variable (define (variable? x) (symbol? x)) ; takes two expressions and is #t if ; they are both the same variable (define (same-variable? x y) (and (variable? x) (variable? y) (eq? x y)))

Fall 2008 Programming Development Techniques 9

Sums

; takes two expressions and creates a sum ; with them as the arguments -- sums are ; simply represented as lists (define (make-sum x y) (list '+ x y)) ; returns #t if the argument is a sum ; a sum is a list whose first element is ; the symbol + (define (sum? x) (and (pair? x) (eq? (car x) '+)))

Fall 2008 Programming Development Techniques 10

Sum Selectors

; selectors for a sum retrieve ; the first and second arguments ; to be added (define (sum-arg1 x) (cadr x)) (define (sum-arg2 x) (caddr x))

Fall 2008 Programming Development Techniques 11

Products

; takes two expressions and creates a product ; with them as the arguments -- products are ; simply represented as lists (define (make-product x y) (list '* x y)) ; returns #t if the argument is a product ; a product is a list whose first element is ; the symbol * (define (product? x) (and (pair? x) (eq? (car x) '*)))

Fall 2008 Programming Development Techniques 12

Product Selectors

; selectors for a product retrieve ; the first and second arguments ; to be multiplied (define (product-arg1 x) (cadr x)) (define (product-arg2 x) (caddr x))

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Fall 2008 Programming Development Techniques 13

It works! (sort of)

(define expr (make-product 'x 'y)) expr --> (* x y) (deriv expr 'x) --> (+ (* x 0) (* y 1)) Should be y Need to make simple reductions – use same trick as was used to reduce rational numbers – in the constructor

Fall 2008 Programming Development Techniques 14

A better make-sum

; a new constructor that simplifies the sum a bit (define (make-sum x y) (cond ((and (number? x) (= x 0)) y) ((and (number? y) (= y 0)) x) ((and (number? x) (number? y)) (+ x y)) (else (list '+ x y))))

Fall 2008 Programming Development Techniques 15

A better make-product

; a new constructor that simplifies the product a bit (define (make-product x y) (cond ((or (and (number? x) (= x 0)) (and (number? y) (= y 0))) 0) ((and (number? x) (= x 1)) y) ((and (number? y) (= y 1)) x) ((and (number? x) (number? y)) (* x y)) (else (list '* x y))))

Fall 2008 Programming Development Techniques 16

Still room for improvement

(define expr (make-product 'x 'y)) expr --> (* x y) (deriv expr 'x) --> y but (define expr (make-product 'x 'x)) expr --> (* x x) (deriv expr 'x) --> (+ x x) (* 2 x) would be better

Fall 2008 Programming Development Techniques 17

Always room for improvement

More sophisticated simplifications are done in Reduce, Macsyma, Mathematica Theoretically, no matter how many simplifications we build into the software, there are always more simplifications that can be made